Some members of Congress did voice objections publicly, but not one (that I’m aware of) actually tried to put into motion any kind of lawsuit, legislation, or Congressional action into motion. There are a few reasons for this, and I’ll detail them below:
First, in America, the Republican party is typically more hawkish*; that is, more willing or even eager to think getting involved in a foreign war is justified or wise. So many Republicans didn’t tried to actually prevent Obama from attacking Libya, because their worldview is more accepting of getting involved in foreign wars if America stands to gain by doing so. Democrats are typically more dovish*, but they’re reluctant to oppose a sitting Democratic President because they worry that doing so would be unpopular with their constituents and would weaken Obama on other issues about which they care more.
Second, Libya in general and Qaddafi in particular are pretty damn unpopular in America, and many conservative Americans were perfectly happy to see us act against Qaddafi, who was a pretty bad guy, if maybe not a lot worse than some other third world dictators with whom America is much more closely aligned. Attacking Qaddafi was reasonably popular with bleeding-heart liberals who loved the narrative of poor Muslims overthrowing their oppressive governments to establish liberal democracies because it confirms a belief that most Muslims are basically good people oppressed by governments that support terror and other bad things. So there was actually a significant base of support for supporting Libyan rebels in general.
Thirdly, Congresspeople don’t like to stick their necks out without a good potential upside. Most of the country doesn’t pay close enough attention to parse the fine distinction between someone who tries to stop the President from attacking that bad guy Qaddafi because of a law that Congress passed that may or may not be Constitutional, and someone who tries to stop the President from attacking Qaddafi because they like Qaddafi. They worried that if they raise any objection, they’ll be painted as weak on evil bad men like Qaddafi, or even as Qaddafi sympathizers. Congresspeople are basically always campaigning for their next election, and they don’t want to give anyone an opening to attack them.
So Congress was in a pickle: they don’t like the President violating the War Powers Resolution (which, as I said, may not be Constitutional) but most of them think that attacking Qaddafi, supporting the Libyan rebels, or both are worthy objectives. They think that military action is popular with their constituents, and they don’t want to be attacked in their next election with claims that they’re Qaddafi sympathizers. Now, they could have tried to keep the War Powers Resolution from being weakened by allowing yet another sitting President to violate it with impunity by passing an explicit authorization for military force in Libya, but then they’re on record supporting the war, which will be a big liability if the war goes badly.
Basically, Congress is a bunch of whiny pushovers who were unwilling to vote either to authorize military force or attempt to enforce the War Powers Resolution because they’re more scared of being on the wrong side than excited to be on the right side.
*This is a big generalization and there are exceptions, but these are the broad trends.